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Properties

Label 2-429-13.3-c1-0-4
Degree $2$
Conductor $429$
Sign $0.797 - 0.602i$
Analytic cond. $3.42558$
Root an. cond. $1.85083$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.18 − 2.06i)2-s + (0.5 + 0.866i)3-s + (−1.83 + 3.17i)4-s + 2.23·5-s + (1.18 − 2.06i)6-s + (−1.82 + 3.16i)7-s + 3.95·8-s + (−0.499 + 0.866i)9-s + (−2.65 − 4.60i)10-s + (−0.5 − 0.866i)11-s − 3.66·12-s + (−3.34 + 1.33i)13-s + 8.69·14-s + (1.11 + 1.93i)15-s + (−1.04 − 1.80i)16-s + (−3.59 + 6.22i)17-s + ⋯
L(s)  = 1  + (−0.841 − 1.45i)2-s + (0.288 + 0.499i)3-s + (−0.915 + 1.58i)4-s + 0.999·5-s + (0.485 − 0.841i)6-s + (−0.691 + 1.19i)7-s + 1.39·8-s + (−0.166 + 0.288i)9-s + (−0.840 − 1.45i)10-s + (−0.150 − 0.261i)11-s − 1.05·12-s + (−0.929 + 0.369i)13-s + 2.32·14-s + (0.288 + 0.499i)15-s + (−0.260 − 0.450i)16-s + (−0.871 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.797 - 0.602i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.797 - 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.699671 + 0.234577i\)
\(L(\frac12)\) \(\approx\) \(0.699671 + 0.234577i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (3.34 - 1.33i)T \)
good2 \( 1 + (1.18 + 2.06i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
7 \( 1 + (1.82 - 3.16i)T + (-3.5 - 6.06i)T^{2} \)
17 \( 1 + (3.59 - 6.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.638 - 1.10i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.817 + 1.41i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.24 - 7.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.923T + 31T^{2} \)
37 \( 1 + (1.07 + 1.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.286 + 0.496i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.70 + 4.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + 2.81T + 53T^{2} \)
59 \( 1 + (-6.50 + 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.809 - 1.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.58 - 11.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.51 + 4.35i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.92T + 73T^{2} \)
79 \( 1 - 6.94T + 79T^{2} \)
83 \( 1 - 5.33T + 83T^{2} \)
89 \( 1 + (-9.21 - 15.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.42 - 11.1i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.89531703986856022746474281090, −10.30862682699794821728712038298, −9.523270893012581734793436330164, −8.969350132312642025649926575690, −8.290732667161151719549139908098, −6.49701940446213377376298277289, −5.42095318772230231278142877582, −3.90480385700826166198103218555, −2.63735503552216833058202609477, −2.01015352496313898474807902514, 0.56946456968804804860870232779, 2.58062505722599290509055259250, 4.61167060840821115402046487059, 5.79804033498925580177118308538, 6.74217302284183142132072765882, 7.23278899589864468906509390576, 8.067670886051382446128805762996, 9.375190679807724946304868248789, 9.673038472191357868746823475191, 10.53082598814468986150829143815

Graph of the $Z$-function along the critical line